I wonder…
Structure
- Phrase + (の/ん) + だろうか
- Verb + ( の / ん ) + だろうか
- Noun + (な の /な ん ) + だろうか
Versions without ん/の are also acceptable but sound old-fashioned
[casual・masculine]
I’m a bit curious about the part where versions without ん/の sound old-fashioned. Several of the examples omit the の.
How common is it to omit the の in real life? In what context would you do it?
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I am also curious about this 
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It is not so much that it is old fashioned, than it is just a feature of ん being used very frequently in spoken Japanese, due to it forming a nice sound link between words and phrases. As it also nominalizes the phrase before it (creates a noun-phrase), there is that function as well. Either the form with or without ん/の may be used confidently, without much fear of sounding strange.
What is the difference between のだろうか and かな?
Hey there @Warrie !
They both mean the same thing, but they are used in different situations. かな is used in more casual situations, like texting and casual conversations, while のだろうか is usually used in writing. のだろうか is also casual, but かな sounds more casual, if that makes sense!
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Makes sense, thank you! Maybe this info should be added to the explanations in BP. I had this question a couple of times with words with the same meaning and each time Google helped me out to explain the difference, in this case I just couldn’t find an answer.
Or for example when I give かな as the answer and then I get a ‘let’s try a different grammatical structure here’ reply, change that into ‘let’s make it slightly more formal’ or something like that?